On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions

Autores
Rossi, Julio Daniel; Saintier, Nicolas Bernard Claude
Año de publicación
2016
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We study the limit as p goes to infinity of the first non-zero eigenvalue λp of the p-Laplacian with Neumann boundary conditions in a smooth bounded domain U of Rn. We prove that λ∞:=lim λp1/p=2/diam(U), where diam(U) denotes the diameter of U with respect to the geodesic distance in U. We can think of λ∞ as the first eigenvalue of the infinity-Laplacian with Neumann boundary conditions. We also study the regularity of λ∞ as a function of the domain U proving that, under a smooth perturbation Ut of U by diffeomorphisms close to the identity, there holds that λ∞(Ut)=λ∞(U)+O(t). Although λ∞(Ut) is in general not differentiable at t=0, we provide sufficient geometric conditions for its differentiability with an explicit formula for the derivative.
Fil: Rossi, Julio Daniel. Universidad de Alicante; España. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Saintier, Nicolas Bernard Claude. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Materia
infinity laplacian
eigenvalue
shape derivative
neumann boundary condition
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/59878
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/59878
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditionsRossi, Julio DanielSaintier, Nicolas Bernard Claudeinfinity laplacianeigenvalueshape derivativeneumann boundary conditionhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We study the limit as p goes to infinity of the first non-zero eigenvalue λp of the p-Laplacian with Neumann boundary conditions in a smooth bounded domain U of Rn. We prove that λ∞:=lim λp1/p=2/diam(U), where diam(U) denotes the diameter of U with respect to the geodesic distance in U. We can think of λ∞ as the first eigenvalue of the infinity-Laplacian with Neumann boundary conditions. We also study the regularity of λ∞ as a function of the domain U proving that, under a smooth perturbation Ut of U by diffeomorphisms close to the identity, there holds that λ∞(Ut)=λ∞(U)+O(t). Although λ∞(Ut) is in general not differentiable at t=0, we provide sufficient geometric conditions for its differentiability with an explicit formula for the derivative.Fil: Rossi, Julio Daniel. Universidad de Alicante; España. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Saintier, Nicolas Bernard Claude. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaUniversity of Houston2016-06info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/59878Rossi, Julio Daniel; Saintier, Nicolas Bernard Claude; On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions; University of Houston; Houston Journal Of Mathematics; 42; 2; 6-2016; 613-6350362-1588CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.math.uh.edu/~hjm/Vol42-2.htmlinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-11-12T09:52:41Zoai:ri.conicet.gov.ar:11336/59878instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-11-12 09:52:42.02CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions
title On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions
spellingShingle On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions
Rossi, Julio Daniel
infinity laplacian
eigenvalue
shape derivative
neumann boundary condition
title_short On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions
title_full On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions
title_fullStr On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions
title_full_unstemmed On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions
title_sort On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions
dc.creator.none.fl_str_mv Rossi, Julio Daniel
Saintier, Nicolas Bernard Claude
author Rossi, Julio Daniel
author_facet Rossi, Julio Daniel
Saintier, Nicolas Bernard Claude
author_role author
author2 Saintier, Nicolas Bernard Claude
author2_role author
dc.subject.none.fl_str_mv infinity laplacian
eigenvalue
shape derivative
neumann boundary condition
topic infinity laplacian
eigenvalue
shape derivative
neumann boundary condition
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We study the limit as p goes to infinity of the first non-zero eigenvalue λp of the p-Laplacian with Neumann boundary conditions in a smooth bounded domain U of Rn. We prove that λ∞:=lim λp1/p=2/diam(U), where diam(U) denotes the diameter of U with respect to the geodesic distance in U. We can think of λ∞ as the first eigenvalue of the infinity-Laplacian with Neumann boundary conditions. We also study the regularity of λ∞ as a function of the domain U proving that, under a smooth perturbation Ut of U by diffeomorphisms close to the identity, there holds that λ∞(Ut)=λ∞(U)+O(t). Although λ∞(Ut) is in general not differentiable at t=0, we provide sufficient geometric conditions for its differentiability with an explicit formula for the derivative.
Fil: Rossi, Julio Daniel. Universidad de Alicante; España. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Saintier, Nicolas Bernard Claude. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
description We study the limit as p goes to infinity of the first non-zero eigenvalue λp of the p-Laplacian with Neumann boundary conditions in a smooth bounded domain U of Rn. We prove that λ∞:=lim λp1/p=2/diam(U), where diam(U) denotes the diameter of U with respect to the geodesic distance in U. We can think of λ∞ as the first eigenvalue of the infinity-Laplacian with Neumann boundary conditions. We also study the regularity of λ∞ as a function of the domain U proving that, under a smooth perturbation Ut of U by diffeomorphisms close to the identity, there holds that λ∞(Ut)=λ∞(U)+O(t). Although λ∞(Ut) is in general not differentiable at t=0, we provide sufficient geometric conditions for its differentiability with an explicit formula for the derivative.
publishDate 2016
dc.date.none.fl_str_mv 2016-06
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/59878
Rossi, Julio Daniel; Saintier, Nicolas Bernard Claude; On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions; University of Houston; Houston Journal Of Mathematics; 42; 2; 6-2016; 613-635
0362-1588
CONICET Digital
CONICET
url http://hdl.handle.net/11336/59878
identifier_str_mv Rossi, Julio Daniel; Saintier, Nicolas Bernard Claude; On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions; University of Houston; Houston Journal Of Mathematics; 42; 2; 6-2016; 613-635
0362-1588
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://www.math.uh.edu/~hjm/Vol42-2.html
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv University of Houston
publisher.none.fl_str_mv University of Houston
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
_version_ 1848598191192145920
score 12.976206

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